Function Repository Resource:

Give the adjugate matrix of a square matrix

Contributed by: Daniel Lichtblau
 ResourceFunction["Adjugate"][mat] gives the transpose of the cofactor matrix of mat.

## Details

The matrix product of a matrix m with its adjugate is diagonal, with all diagonal entries equal to the determinant of m.

## Examples

### Basic Examples (1)

Form the adjugate of a matrix of dimension 2:

 In:= Out= ### Scope (8)

The adjugate is a square matrix of the same dimension:

 In:= In:= Out= In:= Out= The product of the adjugate with the matrix is a diagonal matrix with the same values on the diagonal:

 In:= Out= The values on the diagonal are the determinant of the matrix:

 In:= Out= When a matrix is invertible, the adjugate divided by the determinant gives the inverse:

 In:= Out= The adjugate is defined for symbolic as well as numeric matrices:

 In:= Out= A square matrix need not be invertible:

 In:= This matrix is not invertible because it does not have full rank:

 In:= Out= The adjugate is defined even when the inverse does not exist:

 In:= Out= ### Properties and Relations (2)

The adjugate satisfies a simple identity:

 In:= In:= Out= The adjugate and the matrix of minors are the same up to reordering and signs:

 In:= Out= In:= Out= In:= Out= The adjugate and the matrix of cofactors (implemented as the resource function CofactorMatrix) are transposes of one another:

 In:= Out= ## Requirements

Wolfram Language 11.3 (March 2018) or above

## Version History

• 1.0.1 – 13 May 2021
• 1.0.0 – 14 February 2019